MX4083: MEASURE THEORY (2018-2019)

1. Lebesgue extension of a measure;Algebras and σ-algebras of sets;Measure spaces;Properties of the Lebesgue measure;
2. Lebesgue extension of a measure defined on a ring of sets;
3. σ-additive and σ-semi-additive measures;
4. Extension a measure from a semi-ring of sets to the corresponding ring of sets;
5. Measurable functions;Convergence almost everywhere;
6. Measurable functions and uniform convergence; the Egorov theorem;
7. Definition and properties of measurable functions;
8. Lebesgue integral;The definition and the properties of the Lebesgue integral;The Lebesgue, the Levi and the Fatou convergence theorems for the Lebesgue integral;
9. Comparison of the Lebesgue integral and of the Riemann integral.
10. Absolute continuity and σ-additivity of the Lebesgue integral; the Chebyshev inequality;
11. Lebesgue integral for simple functions;

Syllabus

• Lebesgue extension of a measure;

          Extension a measure from a semi-ring of sets to the corresponding ring of sets;

          Algebras and σ-algebras of sets;

          σ-additive and σ-semi-additive measures;

          Measure spaces;

          Lebesgue extension of a measure defined on a ring of sets;

          Properties of the Lebesgue measure;

• Measurable functions;

          Definition and properties of measurable functions;

          Convergence almost everywhere;

          Measurable functions and uniform convergence; the Egorov theorem;

• Lebesgue integral;

           Lebesgue integral for simple functions;

          The definition and the properties of the Lebesgue integral;

          Absolute continuity and σ-additivity of the Lebesgue integral; the Chebyshev inequality;

          The Lebesgue, the Levi and the Fatou convergence theorems for the Lebesgue integral;

          Comparison of the Lebesgue integral and of the Riemann integral.

Course Aims

The aim of the course is to introduce the basic ideas and techniques of measure theory and Lebesgue integration.

Learning Objectives

By the end of the revision period, students should be able (i) to state and illustrate the definitions of the concepts introduced in the course, (ii) to state the theorems of the course, to explain their significance, and to give examples to indicate the role of the hypotheses, (iii) to demonstrate knowledge and understanding of proof techniques used in the course, (iv) to use the methods and results of the course to solve problems at levels similar to those seen in the course.

In particular, students should be able
- to define a &sigma -algebra, a measure and a measurable function, to check the definitions in examples and to prove simple results (seen and unseen) about these concepts;
- to integrate (in turn) a simple positive measurable function, a positive measurable function and a general L1 function and derive properties of integrals using this path;
- to evaluate integrals of continuous functions on the real line and to evaluate integrals of sequences with respect to counting measure on the positive integers;
- to state, prove and use the Monotone Convergence Theorem, Fatou’s Lemma and the Dominated Convergence Theorem.

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%). Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.